M

3

4

5

6

7

8

9

A10

oswto

11

12

13

14

15

16

©17

K18

19

120

21

g22

t23

m24

w25

d26

f27

m28

29

w30

n31

a32

b33

a34

Q1

nf

(

1 02 d

RO

OF

ARTICLE IN PRESS+ModelSA 23429 1–14

Materials Science and Engineering A xxx (2007) xxx–xxx

On the incorporation of length scales associated with pearlitic andbainitic microstructures into a visco-plastic self-consistent model

D. Canadinc a,∗,1, H. Sehitoglu a,1, H.J. Maier b,2, P. Kurath a,1

a University of Illinois at Urbana-Champaign, Department of Mechanical Science and Engineering,1206 W. Green Street, Urbana, IL 61801, United States

b University of Paderborn, Lehrstuhl f. Werkstoffkunde, D-33095 Paderborn, Germany

Received 17 July 2007; accepted 22 August 2007

bstract

A visco-plastic self-consistent model was modified to account for strain hardening in steels with pearlitic and bainitic microstructures. Occurrencef slip and the contribution of secondary phases in body-centered cubic steels to the strengthening were simulated for tension, compression andhear loadings. The initial texture and texture evolution at larger strains for bcc polycrystalline steels with intrinsic barriers to dislocation motion

D P

ere studied under quasi-static conditions. The crystallographic texture of rails undergoing large-shear strains was closely examined and comparedo the simulations. The results clearly demonstrate that the method has the capability to predict texture evolution in deformed rails. The methodologyffers opportunities for improved materials characterization of various steel microstructures.

2007 Published by Elsevier B.V.

hanic

b 35

c 36

m 37

38

o 39

c 40

S 41

o 42

t 43

m 44

i 45

p 46

o 47

RE

CTE

eywords: Crystal plasticity; Anisotropic material; Constitutive behavior; Mec

. Background and motivation

Steels undoubtedly constitute the most extensively studiedroup of metallic materials to date, and consequently, the litera-ure offers a wealth of information on various aspects, including

odeling of their mechanical response. Despite the existingealth of knowledge on the subject matter, steels continue toemand attention as new applications that call for improved per-ormance warrant a deeper understanding of the strengtheningechanisms.Within the last three decades, many numerical simulations

ere carried out investigating the finite deformation mecha-isms, namely slip and twinning, plastic deformation of steels,

UN

CO

R

Please cite this article in press as: D. Canadinc, et al., Mater. Sci. Eng. A (

nd its relationship with texture. Nevertheless, to the authors’est knowledge, a model has not been forwarded to date thatccounts for the different phases in pearlitic and bainitic steels

∗ Corresponding author. Current address: Department of Mechanical Engi-eering, Koc University, Istanbul, Turkey. Tel.: +1 217 333 1176;ax: +1 217 244 6534.

E-mail addresses: demircan.canadinc@gmail.com, dcanadinc@ku.edu.trD. Canadinc).1 Tel.: +1 217 333 1176; fax: +1 217 244 6534.2 Tel.: +49 5251 60 3855; fax: +49 5251 60 3854.

a 48

s 49

r 50

p 51

ibt

a

921-5093/$ – see front matter © 2007 Published by Elsevier B.V.oi:10.1016/j.msea.2007.08.049

al testing; Visco-plastic self-consistent algorithm

y incorporating the length scales of obstacles specific to theorresponding microstructure in a simple crystal plasticity for-ulation.With this motivation, the present study was undertaken, in

rder to shed more light onto the occurrence of glide in body-entered cubic (bcc) steels, especially at large-plastic strains.pecifically, the present study focuses on the developmentf a modified crystal plasticity model that correctly predictshe deformation response by incorporating the role of fine

icrostructural features of the two-phase bcc steels. The focuss placed on two classes of rail steels, featuring bainitic andearlitic microstructures. The J6 bainitic steel3 is a newly devel-ped high-strength steel [1,2], and possesses a bainitic structure,nd a hardness of 44 on the Rockwell C (HRC) scale. Itstrength is dictated by fine precipitates. The premium rail mate-ial, pearlitic steel, has a hardness of 38 HRC [3] and resistslastic deformation due to the fine pearlite lamellae present-

2007), doi:10.1016/j.msea.2007.08.049

ng an obstacle to dislocation motion. The J6 bainitic steel has 52

een tested on track by TTCI (Transportation Technology Cen- 53

er Inc., Pueblo Co.) under various environmental and loading 54

3 This steel was developed by Transportation Technology Center Inc. (TTCI),subdivision of Association of American Railroads (AAR).

ED

IN+ModelM

2 ce and

c55

w56

m57

fi58

C59

i60

t61

62

d63

s64

(65

e66

m67

a68

s69

o70

t71

d72

s73

a74

fi75

76

e77

m78

b79

p80

o81

m82

r83

a84

t85

t86

n87

s88

p89

t90

p91

b92

d93

b94

w95

r96

s97

98

b99

t100

m101

a102

a103

c104

T105

d106

g107

s108

a109

p110

n111

T 112

m 113

s 114

115

p 116

b 117

s 118

t 119

r 120

i 121

a 122

s 123

i 124

a 125

c 126

i 127

t 128

o 129

l 130

l 131

s 132

v 133

p 134

t 135

p 136

2 137

138

b 139

( 140

a 141

y 142

P 143

s 144

i 145

f 146

t 147

( 148

b 149

( 150

s 151

a 152

153

d 154

d 155

r 156

w 157

o 158

s 159

i 160

as it represents the simplest experimental method of obtaining 161

simple shear deformation. Such tests are very useful for obtain- 162

NC

OR

RE

CT

ARTICLESA 23429 1–14

D. Canadinc et al. / Materials Scien

onditions, and the observations indicated a similar or reducedear and fatigue performance compared to that of the afore-entioned pearlitic steel [2,4], prompting a closer look at thenite deformation mechanisms at large-plastic strains. The 1%rail steel is another pearlitic rail steel, which was considered

n the current work for the verification of the modeling approachaken.

Another contribution of the work presented herein is theevelopment of a methodology to obtain information ontress–strain-texture states of steels at large-plastic strainsexceeding 1.0) beyond the capabilities of standard laboratoryxperiments. The modified visco-plastic self-consistent (VPSC)odel that accounts for the specific details of the bainitic

nd pearlitic microstructures was used to predict the sheartrain hardening response and yield strength in shear, based onnly the materials’ response under uniaxial tensile loading andheir initial textures. The objective was to obtain a material’seformation response and the corresponding texture evolutionimultaneously through a single VPSC simulation, eliminatingdditional experimental observations that are both time wise andnancially demanding in comparison to numerical simulations.

The proposed method utilizes crystal plasticity conceptsstablishing the link between deformation mechanisms at theicrostructural level, and mechanical properties. The link

etween the tensile deformation mode, and the shear and com-ressive responses is established by considering the anisotropyf the materials. This requires the incorporation of deformationechanisms of the materials under investigation. All three mate-

ials considered in this study (J6 bainitic steel, pearlitic steel,nd 1% C rail steel) possess a bcc lattice structure with differentypes of strengthening secondary phases, which alone requireshe theoretical consideration of the complexity of the slip mecha-ism related to the bcc structure. Experimental observations havehown that slip takes place on the {1 1 0}, {1 1 2} and {1 2 3}lanes and in the 〈1 1 1〉direction in bcc crystals [5,6]. Of thesehe {1 1 0} plane is the most nearly close-packed, making it therimary slip plane in bcc materials. As mentioned previouslyy Havner [5], one noteworthy observation regarding the finiteeformation of bcc crystals is that there are few works on slip incc crystals since Taylor and Elam’s early works [7–9]. In thisork, we provide simulations over a large-strain regime for two

ather different microstructures allowing slip on {1 1 0} basedystems.

Similar to slip, texture evolution is another topic that has noteen extensively studied in bcc materials. A large fraction of theexture literature deals with rolling of face-centered cubic (fcc)

etals [10]. Nevertheless, the literature contains studies thatttempted to investigate the relationship between slip activitynd texture evolution in bcc materials [11–13], most of whichoncentrate on the role of pencil glide on the texture evolution.he work of Rollett and Kocks [11] demonstrated that the textureevelopment in tension was essentially the same as for restrictedlide (the {1 1 0}〈1 1 1〉 slip), whereas significant differences

U

Please cite this article in press as: D. Canadinc, et al., Mater. Sci. Eng. A

urfaced in plane strain compression. In the light of this finding,ll possible combinations of slip activity were allowed in thereliminary simulations, yet no drastic differences were revealedeither in the deformation response nor the texture evolution.

i

PR

OO

F

PRESSEngineering A xxx (2007) xxx–xxx

hus, in the current work, only primary slip is considered as theeans of plastic deformation, and therefore the 12 {1 1 0}〈1 1 1〉

lip systems were allowed in the corresponding simulations.In this study two unique strain hardening formulations are

roposed that account for the fine features of pearlitic andainitic microstructures, and simulations are presented fortress–strain response and texture evolution in bcc crystals. Theexture evolution in bcc steels at strains within the experimentalange of standard laboratory equipment and at higher strains isnvestigated to illustrate the trends. Texture evolution is utilizeds a tool for determining the state of stress in rails taken fromervice in an inverse fashion. Specifically, the changes in normal-zed texture intensities of specific crystallographic orientationsre monitored. It is also shown that the use of an isotropic yieldriterion, such as the von Mises criterion (as commonly adoptedn finite element codes), has limitations where the degree ofexture increases with the applied deformation. The anisotropyf a material is manifested through the “Taylor factor” calcu-ated. Our proposed approach incorporates volume fraction ofamellae, and its orientation change upon deformation at largetrains. The transmission electron microscopy observations pro-ide a measure of the length scales associated with the secondaryhases. The work provides improved understanding of deforma-ion in bcc materials where texture evolution is evident in theresence of a secondary phase.

. Experimental techniques

Three different rail steels were investigated in this study:ainitic steel (BS), pearlitic steel (PS) and 1% C rail steel1CRS). The bainitic steel has 0.26% C, 2.0% Mn, 1.81% Si,nd 1.93% Cr, while the ladle analysis for both pearlitic steelsielded 0.91% Mn, 0.66% Si, 0.47% Cr, and 0.79% C for theS and 1.0% C for the 1CRS.4 All three materials have a bcctructure at room temperature. The specimens for all three load-ng scenarios (tension, compression and torsion) were extractedrom rail heads, such that the loading axis of each sample onhe test frame coincides with the rolling direction of the railsFig. 1). Small-scale samples were utilized for tension (dog-one shaped, with a gauge length of 10 mm) and compressionprismatic, with a cross-section of 16 mm2) tests, whereas large-cale tubular specimens (with a gauge section of 30 mm in lengthnd a wall thickness of 1.4 mm) were used for pure torsion tests.

Uniaxial tension and compression experiments were con-ucted as the first step in mechanical testing with the purpose ofefining the fundamental macroscopic properties of the mate-ials. The strain hardening response is also closely monitoredith the aid of these tests. However, owing to the small amountf ductility in tensile tests and relatively narrower strain spantudied, experiments under compression, as well as shear exper-ments, were also carried out. The torsion test was conducted

(2007), doi:10.1016/j.msea.2007.08.049

ng strain hardening data for large-plastic strains (exceeding 1.0) 163

4 The compositions provided reflect weight percentage.

D

ARTICLE IN PRESS+ModelMSA 23429 1–14

D. Canadinc et al. / Materials Science and Engineering A xxx (2007) xxx–xxx 3

F rails ind ecim

a164

p165

t166

167

n168

c169

t170

T171

h172

u173

s174

s175

176

t177

t178

w179

a180

[181

s182

o183

t184

i185

fi186

d187

e 188

e 189

t 190

191

i 192

i 193

c 194

t 195

c 196

u 197

198

d 199

w 200

a 201

t 202

s 203

t 204

m 205

B 206

i 207

TE

ig. 1. All the specimens reported on in the present study were extracted fromirection (i.e. the longitudinal direction of the rail). Only extraction of tensile sp

voiding the friction effects at interfaces between specimens andlatens that occur in compression testing [10] or early facture inension experiments.

The experiments were conducted at quasi-static strain rates,amely at an initial rate of 4 × 10−4 s−1. The strain rate washosen with the purpose of establishing the mechanical proper-ies where strain rate dependence would not be pronounced [14].ests for each loading type were checked for repeatability. Servoydraulic load frames equipped with digital controllers weretilized in conducting the mechanical tests. Miniature exten-ometers were used to accurately measure the strain on theub-sized tensile and compressive samples.

The textural changes were monitored through X-ray diffrac-ion (XRD). The experimental data obtained from XRD or theexture information obtained as an output from the simulationsere interpreted with the aid of the Preferred Orientation Pack-

ge (popLA) developed at the Los Alamos National Laboratory10]. For each case, at least three experimental pole figure mea-urements were taken with XRD as this is the minimum numberf different poles to construct an orientation distribution func-

UN

CO

RR

EC

Please cite this article in press as: D. Canadinc, et al., Mater. Sci. Eng. A (

ion (ODF). ODF is the most complete description of a texturen three-dimensional space. As a general rule, three or more polegure data points are needed that correspond to each orientationistribution (OD) cell computed by discrete methods [10]. Our

i

ft

Fig. 2. Representative TEM images showing the initial microstructu

PR

OO

F

such a way that their loading axis on the test frame coincides with the rollingens is shown. The rail head is shown from the top.

xperimental data was obtained through three measurements forach sample: (1 1 0), (2 0 0), and (2 1 1) poles. These are the poleshat exhibited the peak intensities during the initial θ–2θ scan.

The ODFs for the experimentally measured textures andnverse pole figures were calculated using popLA. The numer-cally predicted textures, on the other hand, were given by theurrent model in the form of Euler angles that represent the tex-ure in the form of orientations assumed by individual grains. Theorresponding ODFs and inverse pole figures were calculatedtilizing popLA.

Complimentary microstructural observations were con-ucted through transmission electron microscopy (TEM), whichas utilized to investigate the microstructural mechanisms such

s dislocation activities or slip systems, in addition to showinghe initial microstructural status of the complex industrial alloystudied (Fig. 2). The difference between the microstructures ofhe two-phase alloys studied is evident from the representative

icroscopy images presented in Fig. 2: (a) bainitic structure forS and (b) pearlitic structure for PS. The fine lamellar structure

s evident in the case of PS, and for BS the TEM observations

2007), doi:10.1016/j.msea.2007.08.049

ndicate the presence of fine precipitates. 208

Having the microstructural information as a background 209

acilitates the construction of the hardening rules. We then place 210

he focus of this study on modeling the macroscopic deforma- 211

re of the materials studied: (a) bainitic and (b) pearlitic steels.

ED

ARTICLE IN+ModelMSA 23429 1–14

4 D. Canadinc et al. / Materials Science and

Fig. 3. Room temperature tensile deformation response and the correspondingsimulation results for BS, PS, and 1CRS. All experiments ended with the failureouu

t212

t213

3214

215

t216

c217

a218

w219

I220

s221

L222

m223

t224

o225

e226

t227

d228

c229

e230

t231

m232

r233

i234

n235

236

i237

t238

a239

b240

τ241

w242

v243

t244

w245

γ246

w 247

s 248

s 249

t 250

r 251

a 252

f 253

ε 254

w 255

t 256

s 257

258

t 259

o 260

E 261

w 262

263

d 264

t 265

ε 266

σ 267

w 268

s 269

s 270

t 271

ε 272

273

M 274

w 275

a 276

277

m 278

m 279

a 280

M 281

282

e 283

p 284

E 285

i 286

e 287

[ 288

NC

OR

RE

CT

f the specimens. The experimental curves represent average of several individ-al experiments carried out for each case. Simulations do not extend beyond theltimate tensile strength levels.

ion response of the materials studied, and the correspondingexture evolution.

. Tensile deformation response and its modeling

The room temperature tensile stress–strain responses of allhree materials studied are reported in Fig. 3. The materials inonsideration exhibit a limited amount of ductility (0.13, 0.12nd 0.095 plastic strain for BS, PS and 1CRS, respectively),hich is not very surprising for this class of high-strength alloys.

n order to model the deformation response of the materialstudied (Fig. 3), the VPSC algorithm originally developed byebensohn and Tome [15] was modified to account for relevanticrostructural details of the different steels and employed as

he numerical tool. This particular algorithm was chosen basedn several reasons. First of all, both literature and our experi-nce have proven the capability of VPSC to effectively modelhe deformation response of high-strength steels with complexeformation mechanisms [14] both in single crystal and poly-rystalline form. More importantly, VPSC captures the texturevolution during deformation and gives the opportunity to assignhe experimentally determined pre-deformation texture of the

aterial as the starting state, which is crucial for obtainingealistic results. The model also enables incorporation of exper-mentally observed additional microstructural features into theumerical algorithm with ease and efficiency [14].

We present the fundamentals of the original VPSC algorithmn order predict deformation response in the plastic regime. Plas-ic deformation occurs when a slip or a twinning system becomesctive. The resolved shear stress, τs

RSS, for a system (s) is giveny

sRSS = ms

iσi (1)

here msi is the vector form of the Schmid tensor and σi is the

ector form of the applied stress. To describe the shear rate inhe system s, a non-linear shear strain rate as a power of τs is

U

Please cite this article in press as: D. Canadinc, et al., Mater. Sci. Eng. A

RSSritten as

˙ s = γ0

(τs

RSS

τs0

)n

= γ0

(ms

iσi

τs0

)n

(2) ρ

PR

OO

F

PRESSEngineering A xxx (2007) xxx–xxx

here γ0 is the reference rate, τs0 the threshold stress corre-

ponding to this reference rate, and n is the inverse of the rateensitivity index. If n is high enough, this description asymp-otically approaches the rate insensitive limit. The total strainate in a crystal can be written as the sum of contributions fromll potentially active systems and can be pseudolinearized asollows [15]:

˙i =[γ0

s∑1

msim

sj

τs0

(ms

kσk

τs0

)n−1]

σj = Mc(sec)ij (σ)σj (3)

here Mc(sec)ij is the secant visco-plastic compliance of the crys-

al which gives the instantaneous relation between stress andtrain rate.

Following Lebensohn and Tome [15], at the polycrystal levelhe same pseudolinear form can be implemented as in the casef Eq. (3) as follows:

˙i = M

(sec)ij (

∑)∑

j+

∑0(4)

here Ei and∑

are the polycrystal strain rate and applied stress.In a continuum that consists of a matrix and inclusions, the

eviations in strain rate and stress between the inclusion andheir overall magnitudes are defined as

˙k = εk − Ek (5)

˜j = σj − Σj (6)

here εk and σj stand for the local (single crystal or grain level)train rate and stress. Utilizing Eshelby’s inhomogeneous inclu-ion formulation one can solve the stress equilibrium equationo derive the following interaction equation [10]:

˜ = −M : σ (7)

The interaction tensor M is defined as

˜ = n′(I − S)−1 : S : M(sec) (8)

here M(sec) is the secant compliance tensor for the polycrystalggregate and S is the visco-plastic Eshelby tensor [10].

The macroscopic secant compliance, M(sec), can be deter-ined by substituting Eqs. (3) and (4) in Eq. (7). Theacroscopic strain rate is evaluated by taking the weighted

verage of crystal strain rates over all the crystals as follows:

(sec) = 〈Mc(sec) : (Mc(sec) + M)−1

: (M(sec) + M)〉 (9)

Iterative solution of Eqs. (3), (7) and (9) gives the stress inach crystal (grain), the crystal’s compliance tensor, and theolycrystal compliance consistent with the applied strain rate

˙i. In this work, we chose the term n (in Eq. (2)) to be in the rate

nsensitive limit (n = 20). As for the interaction equation (8), anffective value of n′ = 1 is used, which ensures a rigid interaction15,16].

(2007), doi:10.1016/j.msea.2007.08.049

The rate of overall dislocation density can be expressed as 289

˙ =∑

n

{k1√

ρ − k2ρ}|γn| (10) 290

DP

RO

OF

ARTICLE IN PRESS+ModelMSA 23429 1–14

D. Canadinc et al. / Materials Science and Engineering A xxx (2007) xxx–xxx 5

F etweeE

w291

a292

293

f294

τ295

w296

e297

a298

r299

a300

τ301

302

τ303

304

s305

√306

307

s 308

τ 309

310

i 311

n 312

τ 313

w 314

t 315

t 316

l 317

t 318

g 319

τ 320

w 321

p 322

e 323

NC

OR

RE

CTE

ig. 4. Schematic showing the incorporation of the contribution of interaction bq. (20) for details.

here k1 and k2 are the geometric constants that define thethermal (statistical) storage of the moving dislocations [10].

We define the flow stress τ in the traditional Taylor hardeningormat as

− τ0 = αμb√

ρ (11)

here α is the dislocation interaction parameter and τ0 is a refer-nce strength, which is related to deformation at the grain levels will be detailed later on. From Eq. (11), with τ0 constant, theate of the flow stress is obtained by taking the time derivatives

˙ = αμbρ

2√

ρ(12)

Substituting Eq. (10) into Eq. (12) results in

˙ =∑

n

{k1

αμb

2− k2

αμb

2√

ρ

}|γn| (13)

From Eq. (11), the following identity is obtained for the

U

Please cite this article in press as: D. Canadinc, et al., Mater. Sci. Eng. A (

quare root of the density of dislocations:

ρ = τ − τ0

αμb(14)

aeps

n pearlite lamellae and glide dislocations into the Voce hardening scheme. See

Once Eq. (14) is substituted into Eq. (13), the rate of flowtress evolution is given by:

˙ =∑

n

{k1

αμb

2− k2

τ − τ0

2

}|γn| (15)

One should note that the term {(αμb/2)k1 − ((τ − τ0)/2)k2}n Eq. (15) is the linear Voce hardening term (Eq. (17)). Havingoted this, Eq. (15) can also be expressed as [10,16]:

˙ =∑

n

{θ0

(τs − τ

τs − τ0

)}|γn| (16)

here θ0 is the constant strain hardening rate, and τs representshe saturation stress in the absence of geometric effects, or thehreshold stress. The hardening is defined by an extended Voceaw [10,14,16], which is characterized by the evolution of thehreshold stress (τs) with accumulated shear strain (Γ ) in eachrain of the form:

s = τ0 + (τ1 + θ1Γ )

(1 − exp

(−θ0Γ

τ1

))(17)

here τ0 is the reference strength, and τ1, θ0 and θ1 are thearameters that define the hardening behavior [10,16]. The hard-ning law defined by Eq. (17) characterizes the onset of plasticity

2007), doi:10.1016/j.msea.2007.08.049

nd the saturation of threshold stress at larger strains. The gen- 324

ral conditions of θ0 ≥ θ1 ≥ 0 and τ1 ≥ 0 need to be fulfilled for 325

ositive hardening, whereas θ0 ≤ θ1 ≤ 0 and τ1 ≤ 0 should be 326

atisfied simultaneously for describing negative hardening. The 327

ED

IN+ModelM

6 ce and

l328

t329

330

a331

t332

s333

a334

i335

a336

i337

3338

V339

340

b341

V342

w343

t344

b345

d346

a347

o348

r349

m350

351

t352

c353

s354

s355

s356

d357

d358

t359

l360

t361

a362

d363

364

s365

i366

t367

w368

t369

m370

a371

g372

373

t374

h375

S376

f377

f378

M379

o380

t381

t382

a 383

d 384

t 385

g 386

387

t 388

e 389

d 390

r 391

ρ 392

w 393

v 394

s 395

( 396

t 397

d 398

e 399

a 400

r 401

l 402

t 403

d 404

i 405

t 406

t 407

b 408

s 409

v 410

o 411

u 412

o 413

s 414

o 415

a 416

p 417

a 418

A 419

f 420

s 421

p 422

t 423

τ 424

w 425

E 426

s 427

T 428

o 429

NC

OR

RE

CT

ARTICLESA 23429 1–14

D. Canadinc et al. / Materials Scien

imit case of the Voce hardening law, namely linear hardening,akes place when τs

1 = 0.VPSC follows the algorithm outlined in Eq. (1) through (17)

nd solves for the stresses corresponding to the given strainshroughout the deformation. The self-consistent algorithm isolved with three nested iterations [16]. Briefly, the outer iter-tion varies the stress and the compliance in each grain. Thentermediate iteration varies the overall tangent modulus of theggregate (see Ref. [17] for more details), whereas the innerteration varies the overall secant compliance.

.1. Incorporation of the pearlitic microstructure into thePSC algorithm

At this point, the authors would like to point at two possi-ilities regarding modeling with VPSC. The first is to use thePSC in its original form for carrying out simulations, whichould return satisfactory fits for all cases. However, we note

hat this approach basically means disregarding the differencesetween the microstructures of pearlitic and bainitic steels andoes not account for the variation in microstructural constituents,nd thus, the predictive capabilities would be rather limited. Inrder to present a more realistic approach, we chose to incorpo-ate the relevant microstructural features of pearlitic and bainiticicrostructures into the VPSC algorithm.The pearlitic microstructure possessed by PS and 1CRS fea-

ures cementite lamellae inside the pearlite colonies. Upon initialooling of the steel, in each austenite grain, the pearlite lamellaetarts to grow in a particular direction at a specific nucleationite. Eventually, the grain interiors are occupied by a lamellartructure that is made of “colonies” of lamellae grown in certainirections (Fig. 4). The corresponding orientation of the lamellaeepends on the orientation of the grains and the positioning ofhe grain boundaries, which directly affect the nucleation of theamellae during the initial cooling of the steel. In a microstruc-ure with a (nearly) random texture, however, it is reasonable tossume that the lamellar cells also represent a random orientationistribution over the whole aggregate.

It is very well known that the lamellae (made of Fe3C) con-titute hard microstructural barriers, which have a substantialnfluence on the mechanical properties of pearlitic steels. Inhe current study, we incorporate the length scales associatedith the pearlite lamellae into the Voce hardening formulation

o capture the role of the pearlitic microstructure on the defor-ation response. Specifically, we consider the pearlite lamellae

s impenetrable and to constitute obstacles against dislocationlide on the active slip systems (Fig. 4).

In previous studies [14,18,19] we modeled the contribu-ion of high-density dislocation walls (HDDWs) to the overallardening by utilizing a unique strain hardening formulation.pecifically, we modeled the HDDWs as impenetrable walls thatorm obstacles to dislocation motion, but also evolve in volumeraction as a result of this interaction with glide dislocations.

U

Please cite this article in press as: D. Canadinc, et al., Mater. Sci. Eng. A

oreover, the model also takes into account the reorientationf HDDWs due to plastic deformation. In the current study, wereated pearlite lamellae similar to HDDWs, yet with impor-ant differences. In particular, the pearlite lamellae are modeled

mfao

PR

OO

F

PRESSEngineering A xxx (2007) xxx–xxx

s impenetrable elongated inclusions that reorient in the matrixue to plastic deformation, however, as opposed to HDDWs,hey do not evolve in volume fraction due to interaction withlide dislocations.

The present model utilizes a crystal plasticity description ofhe strain rate at the single crystal level, and the reference stressvolves with the dislocation density. The pearlite lamellae–glideislocation system interaction is incorporated into the overallate of dislocation density, such that Eq. (10) becomes

˙ =∑

n

{k1√

ρ − k2ρ}|γn| +∑

n

∑q

K

dbsin θnq|γn| (18)

here K is a geometric constant, and b represents the burgersector [14,18–20]. The first term

∑n{k1

√ρ − k2ρ}|γn| repre-

ents the athermal (statistical) storage of moving dislocationsk1

√ρ) and dynamic recovery (−k2ρ) [20], whereas the second

erm∑

n

∑q(K/db) sin θnq|γn| accounts for the contribution

ue to pearlite lamellae formed parallel to a plane q acting asffective obstacles to the moving of dislocations gliding on thective slip system n (Fig. 4). In other words, the second termepresents a phenomenological geometric accumulation of dis-ocations at the boundaries of pearlite lamellae, which subdividehe grains (and the lamellar cells within the grains) and therebyecrease the mean free path of dislocations [20]. The angle θnq

s the angle between the direction of slip in the active slip sys-em n and the direction of pearlite lamellae on (or parallel to)he plane q, and is incorporated as a measure of the interactionetween pearlite lamellae and glide dislocations in a geometricense. The angle θnq is a variable, such that it can take differentalues depending on the active slip systems and the orientationf pearlite lamellae they interact with. Moreover, the θnq contin-es to change as the deformation progresses due to the rotationf the lamellae in the matrix. The term d represents the averagepacing between the lamellae. The term γn stands for the ratef shear in the active slip system n. The lamellae are treateds impenetrable barriers to dislocation motion, acting as hardhases in the matrix, similar to precipitates. Accordingly, theyre modeled as (elongated) ellipsoidal inclusions in the matrix.s plastic strain progresses, Orowan loops are stored at the inter-

aces of the pearlite lamellae, giving rise to long-range internaltresses in the matrix [20]. This additional hardening (τB) due toearlite lamellae acting as hard phases in the matrix is includedhrough the term:

B = 2fξμ∑

n

|γn| (19)

here the terms μ, ξ, and f stand for the shear modulus, theshelby accommodation factor for elongated ellipsoidal inclu-ions [14], and the volume fraction of HDDWs, respectively.he

∑nγ

(n) term is simply the summation of the shear strainsn the active slip systems and represents the total matrix mis-

(2007), doi:10.1016/j.msea.2007.08.049

atch strain brought about by the pearlite lamellae. The volume 430

raction of cementite (or impenetrable lamellae) (f) is a constant, 431

nd taken as 0.4 in the current study. Following the incorporation 432

f the additional hardening terms due to the pearlite lamellae, 433

DO

OF

IN PRESS+ModelM

ce and Engineering A xxx (2007) xxx–xxx 7

t434

τ435

436

437

438

t439

a440

a441

F442

p443

e444

w445

a446

d447

w448

N449

t450

[451

[452

g453

e454

a455

a456

t457

t458

l459

460

p461

s462

d463

b464

M465

t466

s467

H468

s469

3470

V471

472

p473

a474

a475

[476

t477

s478

t479

d480

a481

l482

a483

s484

Fs

b 485

s 486

d 487

l 488

i 489

t 490

[ 491

d 492

a 493

494

s 495

a 496

τ 497

w 498

v 499

m 500

501

a 502

τ 503

w 504

r 505

s 506

t 507

a 508

a 509

i 510

NC

OR

RE

CTE

ARTICLESA 23429 1–14

D. Canadinc et al. / Materials Scien

he original rate of flow stress τ defined in Eq. (16) becomes

˙ =∑

n

[α2μ2bK

4d(τ − τ0)

f

1 − f

∑q

sin θnq

+2fξμ +{

θ0

(τs − τ

τs − τ0

)}]|γn| (20)

In order to represent the orientation of pearlite lamellae inhree-dimensional space and account for their geometric inter-ction with the glide dislocations, we assumed that the texturelso reflects the orientation distribution of the pearlite lamellae.or this purpose, the plane q that the pearlite lamellae form on orarallel to was taken as the plane, the normal of which is the ori-ntation vector of the corresponding grain (Fig. 4). Accordingly,e adopt the following convention: the pearlite lamellae are

ssigned a single orientation in each grain, such that the longitu-inal axis of each lamella in the corresponding grain coincidesith or is parallel to a vector v- on plane q with a plane normal

- q. The relationship between v- and N- q is such that they form thewo of the three axes of a local frame. For instance, if N- q is the1 0 0] unit vector in the reference frame, then v- would be in the0 0 1] direction of the reference frame (Fig. 4). Thus, for eachrain, the necessary rotation for the [1 0 0] unit vector of the ref-rence frame to coincide with the surface normal was calculated,nd then applied to the [0 0 1] axis of the reference frame. Thispproach was basically chosen in order to assign orientationso pearlite lamellae in each grain in a consistent manner, andhereby describe the geometric relationship between cementiteamellae and glide dislocations in the reference frame.

Finally, the reorientation of pearlite lamellae was accom-lished by adopting a scheme similar to the twin reorientationcheme in VPSC. Specifically, the pearlite lamellae were rotatedue to plastic deformation within a grain. This was accomplishedy representing the calculated lamellar orientations in terms ofiller indices and having the reorientation scheme determine

he rotation imposed by plastic deformation. In our previoustudies [14,18,19] we successfully accounted for the rotation ofDDWs in Hadfield steel single and polycrystals utilizing the

ame approach.

.2. Incorporation of the bainitic microstructure into thePSC algorithm

In a previous study [20] we successfully modeled the role ofrecipitation hardening on the deformation response of nitrogen-lloyed Hadfield steel single crystals. In the current work, wedopted the strain hardening formulation developed previously20]. In the present approach, we consider the carbides withinhe bainite modifying the critical shear stress on the slip systems,imilar to precipitates (Fig. 5). This is achieved by modifyinghe Voce hardening formulation (Eq. (16)) accounting for theislocation–particle interaction and by considering the particles

U

Please cite this article in press as: D. Canadinc, et al., Mater. Sci. Eng. A (

s one of the controlling factors of the mean free path of dis-ocations. As plastic strain progresses, Orowan loops are storedround the precipitates, which gives rise to long range internaltresses in the matrix. The back stress evolves rapidly at the

τ

wu

PRig. 5. Schematic showing the incorporation of the contribution of particle

trengthening into the Voce hardening scheme. See Eq. (25) for details.

eginning of deformation but tends to saturate with increasingtrain due to plastic relaxation mechanisms around the particlesuring deformation, such as formation of prismatic dislocationoops, and dislocation annihilation. We treat semi-coherent andncoherent particles as elastic inclusions and follow a formula-ion initially similar to Bate et al. [21], Brown and co-workers22–24] and Barlat and Liu [25]. The exact formulation differs inetails as we also incorporate geometric storage of dislocationss a contribution to the forest hardening.

Assuming that the aggregates of carbides are elastic inclu-ions in the matrix, applying a uniform strain to the body that isssociated with a stress, τm, gives the overall stress as

= (1 − f )τm + fτp (21)

here τp is the stress due to the carbides within the bainite, f theolume fraction of precipitates, and τm is the shear stress in theatrix.Considering the Eshelby inclusion problem, τp can be written

s

p = C(I − S)γp (22)

here I and S are the fourth order identity and Eshelby tensors,espectively, C the elastic modulus tensor and γp is the plastictrain discontinuity at the matrix/particle interface. Technically,he value of the strain discontinuity should be less than thepplied strain due to plastic relaxation around the particle aftercertain amount of strain. Eq. (22) can be written for spherical

nclusions as [26]:

2007), doi:10.1016/j.msea.2007.08.049

p = 2μχDγ∗p (23) 511

here χ is the Eshelby accommodation factor, μ the shear mod- 512

lus, D is the modulus correction term and * represents the 513

ED

IN PRESS+ModelM

8 ce and Engineering A xxx (2007) xxx–xxx

u514

d515

m516

517

m518

d519

T520

t521

i522

B523

s524

s525

s526

γ527

w528

t529

v530

(531

t532

f533

τ534

535

I536

c537

t538

t539

c540

[541

t542

3543

544

t545

O546

w547

s548

m549

{550

t551

o552

n553

p554

r555

556

i557

p558

e559

m560

e561

n562

Table 1Voce hardening parameters defining the room temperature deformationresponses of all three materials under uniaxial tensile, torsional and uniaxialcompressive loadings

τ0 (MPa) τ1 (MPa) θ0 (MPa) θ1 (MPa)

BS 373 355 2.0 × 104 233PS 275 413 1.0 × 103 371

563

o 564

y 565

i 566

e 567

o 568

t 569

c 570

t 571

m 572

r 573

o 574

o 575

[ 576

577

y 578

w 579

[ 580

m 581

t 582

s 583

d 584

g 585

[ 586

t 587

o 588

c 589

τ 590

l 591

a 592

τ 593

594

m 595

l 596

p 597

s 598

t 599

e 600

i 601

parameters utilized to predict the room temperature stress–strain 602

NC

OR

RE

CT

ARTICLESA 23429 1–14

D. Canadinc et al. / Materials Scien

nrelaxed portion of the plastic strain. D is orientation depen-ent because of the orientation dependency of the single crystaloduli.The anisotropy due to the precipitates is incorporated into the

odel by the magnitude of the accommodation factor, which isependent upon the shape and orientation of the precipitates.he term fpτp opposes the applied stress during the deforma-

ion, resulting in a back stress. The back stress evolves withncreasing strain and tends to saturate due to plastic relaxation.rown and Stobbs [24] demonstrated that the plastic relaxation

train is a function of the applied strain level, and the precipitateize, shape and volume fraction, and can be approximated forpherical precipitates as

∗p =

(8πb

α2γpr0

)1/8

α

(8γpb

πr0

)1/2

(24)

here r0 is the radius of particles. The first term in Eq. (21) ishe stress in the matrix, which depends upon the internal stateariable, the dislocation density, and can be found by using Eq.16). Replacing Eq. (16) and the time derivative of Eq. (24) intohe time derivative of Eq. (21) results in the constitutive relationor stress evolution with plastic relaxation around particles:

˙=[(

1−2π

3

r2

d2

)θ0

{τs−τ

τs−τ0

}+πχ

r2

d2 μDA1

γ5/8

]∑n

|γn|(25)

n Eq. (25), the term d represents the average distance betweenarbides in the bainite. A is a derived parameter, which is a func-ion of the carbides and the inter-particle distance [20]. Ideally,he volume fraction of particles depends on the radius of parti-les and the distance between particles (Fig. 5), fp = (2π/3)(r/d)2

27]. In this case, we take the volume fraction of the carbides inhe bainite as 0.4.

.3. Simulations

We modeled the room temperature uniaxial tensile deforma-ion response of all three materials under investigation (Fig. 3).nly primary slip, namely slip on the {1 1 0}〈1 1 1〉 system,as considered as the means of plastic deformation during

imulations, as at the temperatures considered iron-based bccaterials tend to deform predominantly by slip activity on

1 1 0} planes [28]. The initial textures measured experimen-ally via XRD were used as an input defining the initial statef the microstructure. Temperature and rate sensitivity wereot taken into account, considering that the deformation takeslace at low homologous (room) temperature and a slow strainate.

The critical step in the modeling of the deformation responses the determination of the hardening parameters that define thelastic flow during straining. To claim any success for the mod-

U

Please cite this article in press as: D. Canadinc, et al., Mater. Sci. Eng. A

ling effort presented herein, the parameters must be physicallyeaningful, in addition to providing a good prediction of the

xperimental data. There are four parameters to be determined,amely the Voce hardening parameters τ0, τ1, θ0 and θ1.

rTfm

PR

OO

F

CRS 356 358 1.1 × 103 239

For a polycrystalline material with grains assuming variousrientations in 3D space, the determination of the microscopicield strength (τ0), or the critical resolved shear stress (CRSS),s not as straightforward as it is for a single crystal, which is gov-rned by the Schmid’s Law. In reality, for each grain, dependingn the orientation it assumes, the effective stress imposed onhat grain will change, bringing about a different yield pointompared to grains oriented differently. An early solution tohis problem was proposed by Sachs, who postulated that theicroscopic yield point of a polycrystalline aggregate could be

epresented by the average of critical resolved shear stressesf all grains in the aggregate [29]. However, this is a ratherversimplifying approach, as pointed out by Taylor, cf. Hosford29].

In the present study, the determination of the microscopicield strength for the polycrystalline aggregate was carried outith the aid of simulations, however, based on Taylor’s analysis

29,30]. The reference strength τ0 for each material was deter-ined such that the values chosen lead to the correct onset of

he plastic regime, and therefore the accurate macroscopic yieldtrength level for the tensile deformation response. In order thisetermination process not to be trivial, we started with an initialuess based on Bishop and Hill’s work [30]. A simple algorithm31] was written based on this work that takes the experimen-ally determined initial textures into account to reflect the levelf anisotropy possessed by each material, and calculates theorresponding Taylor factor. Accordingly, the initial guess for0 is related to the macroscopic yield strength (σy) (in a simi-ar fashion to the Schmid’s Law) through the Taylor factor (M)s

0M = σy (26)

This initial guess of τ0 is modified as necessary to accom-odate the correct predictions of the macroscopic yield strength

evel and the onset of the plastic regime for each material whileredicting the tensile stress–strain response. Once the referencetrength τ0 was determined for each material, the other parame-ers were established by simply trying to predict the (post-yield)xperimental hardening behavior displayed under tensile load-ng, following the guidelines for the Voce hardening law. The

(2007), doi:10.1016/j.msea.2007.08.049

esponse of the three materials under investigation are given in 603

able 1. Utilizing these parameters, the current model success- 604

ully simulated the tensile deformation response for all three 605

aterials (Fig. 3).

D

ARTICLE IN PRESS+ModelMSA 23429 1–14

D. Canadinc et al. / Materials Science and Engineering A xxx (2007) xxx–xxx 9

Fso

4606

i607

608

a609

i610

o611

s612

i613

m614

i615

b616

l617

618

s619

m620

n621

s622

i623

d624

r625

T626

a627

t628

t629

E630

c631

b632

633

m634

d635

t636

g637

r638

a639

d640

i641

d642

t643

t644

t645

Table 2Yield strength in shear: comparison of experimentally determined and numeri-cally predicted values

Yield strength in shear (MPa)

Experimental Predicted

BS 575 568P1

t 646

d 647

648

i 649

d 650

i 651

( 652

i 653

v 654

e 655

5 656

d 657

658

a 659

w 660

l 661

e 662

t 663

m 664

p 665

c 666

F 667

c 668

c 669

onset of the plastic regime still show a good agreement between 670

the simulations and experiments. One should note that the 671

material parameters used for compression simulations were not 672

adjusted. 673

NC

OR

RE

CTE

ig. 6. Room temperature torsional deformation response and the correspondingimulation results for BS, PS, and 1CRS. All experiments ended with the failuref the specimens.

. Predicting the shear response and the yield strengthn shear

The most reliable way of determining the shear response ofmaterial is basically a pure torsion test conducted on a spec-

men of the same material. However, when it comes to a fullr extensive characterization of a new material, many factors,uch as macroscopic properties, microstructure, and texture arenvolved. Experimental investigation of all these aspects not only

eans spending considerable amount of time and effort perform-ng various laboratory tests, but it also restricts the results to theoundaries dictated by the strain values that are attainable in theaboratory.

Utilizing the current model we aim to effectively establishome of the important characteristics of a material based onuch less experimental effort. In what follows, a method for

umerically predicting the shear yield strength in shear andtress–strain response and texture evolution under shear load-ng is outlined. One way of predicting the yield strength andeformation response in shear based on the tensile deformationesponse is to utilize a yield criterion, such as the von Mises andresca yield criteria [32]. However, these yield criteria providequick and good estimate of the shear yield strength based on

he macroscopic tensile yield strength for engineering applica-ions, only under the assumption of the material being isotropic.ven then it has been shown that they fall short of reflecting theorrect form of the yield potential [29,30], and the differencesecome more pronounced when anisotropy is present [30].

An alternative and better way is to relate all different defor-ation modes at the macroscopic level to the microscopic

eformation at the grain (or single crystal) level. At this point,he capability of predicting the deformation response for anyiven deformation relies only on correctly modeling the mate-ial’s response at the microscopic level. This was outlined andchieved in the previous section while modeling the tensileeformation response of the three materials. No matter what themposed macroscopic deformation mode is, at the grain level, the

U

Please cite this article in press as: D. Canadinc, et al., Mater. Sci. Eng. A (

eformation in a single crystal on any slip system is carried outhrough shearing. Therefore, the hardening parameters relatinghe microscopic deformation mechanisms to the macroscopicensile deformation response are expected to successfully cap-

Fstl

PR

OO

F

S 462 450CRS 500 514

ure stress–strain behavior displayed under any other imposedeformation mode.

The shear simulations were carried out based on the harden-ng parameters determined through the modeling of the tensileeformation response (Table 1). A close prediction of the exper-mental shear response (Fig. 6) and the yield strength in shearTable 2) were obtained for all three materials without any mod-fications to the material parameters. The numerically predictedalues matched the experimental ones, with a negligible differ-nce.

. Further validation—predicting the compressiveeformation response

Investigating the correspondence between the simulationsnd experimental data of compressive stress–strain behavioras utilized as a separate tool of validation for the method uti-

ized. As it is possible to predict the shear response based onxperimental tensile stress–strain data, one should also be ableo obtain the compressive response the same way. The current

odel was utilized to check the validity of the method, andredicted the deformation response of all three materials underompressive loading (Fig. 7). Admittedly, the results shown inig. 7 represent not as favorable agreement between numeri-al and experimental data when compared to the previous twoases (tension and torsion). However, the stress levels and the

2007), doi:10.1016/j.msea.2007.08.049

ig. 7. Room temperature compressive deformation response and the corre-ponding simulation results for BS, PS, and 1CRS. All experiments ended withhe failure of the specimens. Simulations were not extended beyond certain strainevels where buckling initiates.

ED

ARTICLE IN PRESS+ModelMSA 23429 1–14

10 D. Canadinc et al. / Materials Science and Engineering A xxx (2007) xxx–xxx

bcc

6674

(675

676

i677

t678

t679

i680

s681

t682

w683

684

i685

t686

t687

a688

t689

t690

p691

692

o693

a694

[695

t696

t697

c698

b699

o700

p701

t702

(703

i704

a705

n706

t707

a708

c709

a710

c711

c712

713

a714

n715

t716

s717

t 718

p 719

p 720

c 721

i 722

( 723

724

a 725

t 726

“ 727

m 728

t 729

t 730

u 731

732

b 733

L 734

s 735

s 736

o 737

p 738

f 739

d 740

t 741

P 742

w 743

( 744

t 745

746

a 747

g 748

w 749

o 750

c 751

t 752

S 753

a 754

o 755

NC

OR

RE

CT

Fig. 8. Stereographic triangle showing the texture evolution in

. A theoretical study: texture evolution at largerhypothetical) strains

The amount of ductility available under tensile loading lim-ts the strain span for which we could experimentally observehe texture evolution due to plastic deformation. Similarly, forhe torsion tests, due to the nature of the shearing process, its hard for a preferred or strong texture to develop [10] unlessignificantly large strains are attained. As for the observation ofhe compressive response, the results are limited to the strainshere instability or buckling initiates.As a part of the effort to characterize the three materials under

nvestigation, the present study was extended to beyond the prac-ical constraints, and the current model was used to explorehe deformation response and accompanying texture evolutiont large-strains typical of rail applications. BS and PS underorsion were considered for this theoretical investigation. Theexture evolution was closely monitored based on the numericalrocedure explained earlier.

With regards to the texture evolution in bcc materials, the-retically, slip can take place on three planes: {1 1 0}, {1 1 2},nd {1 2 3} planes, once a bcc crystal is deformed in tension28]. On all these planes, the slip direction is 〈1 1 1〉, giving riseo a change in texture as depicted in Fig. 8 [5,28], which showshat there is a rotation towards the (1 1 1) pole for the tensionase. Accordingly, one would expect to see more grains of acc polycrystalline aggregate to assume either [1 1 1] or otherrientations in the vicinity of [1 1 1] (Fig. 8) as the deformationrogresses. In the case of compression of a bcc crystal, however,he rotation is reversed, and takes place toward the (1 0 1) poleFig. 8). As for the torsion texture, the rotation of orientationss toward (1 1 1) and (1 0 1) poles, and the (1 1 1)–(1 0 1) bound-ry. In the present case, neither the experimentally determinedor the corresponding numerically calculated deformation tex-ures revealed such information clearly as the strains practicallyttainable did not allow its monitoring. However, such effectsould be relevant for the actual rails in service. Based on thisrgument the authors decided to extend the study to hypotheti-al strains to observe the texture evolution closely as a part ofharacterization of the current materials.

The present model was utilized to compute the textures

U

Please cite this article in press as: D. Canadinc, et al., Mater. Sci. Eng. A

t hypothetical shear strains (0–2.0 in torsion) employing theumerical procedure outlined in the previous sections. The tex-ure evolution was monitored for various strain levels within thepecified limits for the materials under consideration. Although

arcp

PR

OO

F

crystals under tensile [5], compressive, and torsional loading.

he widely used inverse pole figures give an idea of the overallicture of the texture evolution due to deformation, they do notresent a clear understanding of this evolution in the presentase. Therefore, the texture evolution was investigated by mon-toring the change in texture intensity of certain orientations[0 0 1], [1 1 2], [1 1 1] seen in Fig. 8) due to deformation.

With regards to representing the textures, the preference ofcrystallographic orientation over the other(s) or a tendency

owards a pole in a given texture is most easily represented bytexture intensity”. The texture intensity is obtained by the nor-alization of densities of the orientations (or poles). There exist

wo ways of normalization: to a value of 1.0 for the average; oro a value of 1.0 for the integral [10]. The former method wastilized throughout this study.

In three-dimensional space, an orientation can be describedy three angles Ψ , Θ and Φ, which are the Euler angles [29].et (α, β, λ) represent a specific point with α, β and λ corre-ponding to specific values of Ψ , Θ and Φ in an orientationpace represented by the three Euler angles. The total intensityf diffracted X-rays measured at the point (α, β, λ) on an (h k l)ole figure comes from all crystals (grains) in the sample satis-ying the Bragg condition [10]. For the (h k l) pole, the total poleensity from crystals satisfying the Bragg condition is given (inhe normalized form) by

(h k l)(α, β) = 1

2π

∫ 2π

0F (Ψ, Θ, Φ) dΓ (27)

here Γ denotes the path through the orientation distributionOD) corresponding to rotation about the (h k l) pole, and F ishe OD [10].

It was earlier noted that rotation takes place towards (1 1 1)nd (1 0 1) poles, and the (1 1 1)–(1 0 1) boundary in torsion. Theeneral tendency in the current numerical observations agreesith this statement; however, the intensity change of the [1 1 1]rientation is weaker than expected (Fig. 9). The reason for thisontradiction could be twofold: the general trend (Fig. 8) is thathere is a rotation towards the [1 1 1] orientation (or its vicinity).till, the shear strains assumed hypothetically (maximum of 2.0)re below the strain levels needed for the intensity of the [1 1 1]rientation to be pronounced. However, one needs to be careful

(2007), doi:10.1016/j.msea.2007.08.049

s there is no chance of comparing the hypothetical simulation 756

esults with any experimental observations due to the physical 757

onstraints. Therefore the investigation of the capability of the 758

roposed method to capture the texture evolution needs further 759

D

ARTICLE IN+ModelMSA 23429 1–14

D. Canadinc et al. / Materials Science and

Fig. 9. The “low” and “high” rail textures of (a) BS and (b) PS in comparisonwith hypothetical torsion simulation results and experimental observations. Thefracture shear strain from torsion experiments for both BS (0.57) and PS (0.61)at

e760

h761

762

t763

b764

T765

t766

o767

s768

a769

m770

r771

d772

773

o774

t775

w776

(777

778

t779

c780

t781

w782

s783

a784

d785

786

e 787

v 788

c 789

t 790

o 791

l 792

o 793

l 794

m 795

c 796

b 797

b 798

i 799

w 800

801

a 802

c 803

f 804

A 805

c 806

w 807

j 808

t 809

V 810

m 811

i 812

l 813

a 814

f 815

n 816

s 817

a 818

i 819

d 820

w 821

e 822

second decimal only. The simulation results also agreed well 823

with the stereographic maps of Taylor factors tabulated in Hos- 824

ford’s work [29] considering the change of texture with ongoing 825

deformation. 826

UN

CO

RR

EC

TE

re shown along with the corresponding shear strains for low- and high-railextures of both materials: 1.00 and 1.10 for BS, and 0.95 and 1.10 for PS.

laboration, and is well beyond the scope of the work presentederein.

In the real case, where the wheel rolls and slides on the rail,he deformation and the accompanying texture evolution arerought about mainly by shear applied to the surface of the rail.o be able to see where the worn rails stand in terms of texture,

he textures of rails taken from service were compared to thenes corresponding to experimental and hypothetical loadingcenarios. For this purpose, samples extracted from the “low”nd “high” rails of BS and PS underwent XRD. The texture waseasured near the contact surface (from samples extracted from

ails in service), and the data shown was computed for the rollingirection.

The “low rail” is the inner, and the “high rail” is the outer railf a curved track, and normally, the high-rail wears more thanhe low rail. Both J6 bainitic and pearlitic steel low and high railsere taken from the test loop with high tonnage: 450 megatonnes

MGT) for BS, and 517 MGT for PS.The comparison between the experimentally measured tex-

ures of the rails and the hypothetical simulation textures wasarried out in terms of texture intensities, as the latter reflecthe texture evolution in a larger strain span. The comparisonas made utilizing the torsion textures, which reflect the simple

Please cite this article in press as: D. Canadinc, et al., Mater. Sci. Eng. A (

hear response of the material. Simple shear could be considereds a “close representation” of the real case, where shear is theominant deformation mode (Fig. 9).

Fis

PR

OO

F

PRESSEngineering A xxx (2007) xxx–xxx 11

The intensities of (1 1 1), (0 0 1) and (1 1 2) poles from thexperimentally determined textures of the rails taken from ser-ice were matched with a shear strain level, such that theyoincide with the hypothetical simulation results. Accordingly,he texture of BS low rail corresponds to a shear strain levelf 1.00, and the BS high-rail texture intensities are equiva-ent to those of hypothetical simulation results at a shear strainf 1.10. Both of these values are well beyond the practicalimit of 0.57 shear strain achieved in laboratory tests for this

aterial. Similarly, the low- and high-rail textures of the PSorrespond to shear strains of 0.95 and 1.10, which is againeyond the limit 0.61 attainable in the laboratory. The differenceetween the low and high rails of both materials is noticeable,ndicating the evolution of texture with increasing amount ofear.The change of level of anisotropy was also closely monitored

t larger strains. To reflect this change, the Taylor factors werealculated for a polycrystalline aggregate under tensile loadingollowing various levels of imposed simple shear deformation.t the shear strain levels for which the texture evolution was

losely monitored (Fig. 9) the corresponding simulation texturesere taken as the initial state of a polycrystalline aggregate sub-

ected to uniaxial tension. The Taylor factors were calculatedhrough Eq. (18) relating the yield strength levels obtained fromPSC simulations to the reference strength τ0 that reflects theicroscopic yield strength (Table 1). The results point out to an

ncreasing Taylor factor (and therefore level of anisotropy) atarger strains (Figs. 10 and 11). Even at the initial state, prior tony shearing, for both BS and PS, the Taylor factors are differentrom that of a random polycrystalline aggregate, indicating theeed for an anisotropic yield criterion for predicting the yieldtrength in shear. This need becomes more pronounced as themount of previously imposed shear deformation on a materialncreases (Figs. 10 and 11). In order to verify the analysis con-ucted, the algorithm based on Bishop and Hill’s work [30,31]as applied to the simulation textures at the shear strain lev-

ls of interest. The Taylor factors obtained were different to a

2007), doi:10.1016/j.msea.2007.08.049

ig. 10. Taylor factors (M) for BS obtained from simulations. Correspondingnverse pole figures for the undeformed state and predicted ones for 2.0 sheartrain are shown. The degree of anisotropy increases with further deformation.

ED

ARTICLE IN+ModelMSA 23429 1–14

12 D. Canadinc et al. / Materials Science and

Fis

7827

7828

829

t830

s831

r832

i833

s834

s835

u836

t837

t838

t839

t840

841

s842

t843

l844

v845

p846

p847

7848

849

e850

m851

r852

t853

r854

e855

n856

T857

m858

g859

860

t861

a862

t863

p 864

T 865

c 866

u 867

868

r 869

o 870

a 871

b 872

t 873

i 874

t 875

a 876

a 877

e 878

t 879

i 880

881

s 882

( 883

t 884

a 885

3 886

w 887

t 888

b 889

g 890

b 891

l 892

s 893

4 894

r 895

t 896

g 897

a 898

m 899

3 900

7 901

In the simulations, the Voce hardening rule was modified to 902

account for the features of the pearlitic and bainitic microstruc- 903

tures, and then utilized to model the deformation response of 904

NC

OR

RE

CT

ig. 11. Taylor factors (M) for PS obtained from simulations. Correspondingnverse pole figures for the undeformed state and predicted ones for 2.0 sheartrain are shown.

. Final remarks and discussion of results

.1. Texture evolution

The work demonstrated that the experimentally measuredextures of the worn rails could be matched to those of torsionimulations extending beyond the practical limits of the labo-atory tests. This confirms that computer simulations providensight where experimental data is difficult to obtain. The corre-pondence of the intensities of different poles at the same sheartrain level is another indication of the success of the methodsed in the prediction of shear response utilizing the VPSC. Theexture of the tested samples agreed very well with the simula-ions at the corresponding strains, supporting the idea of utilizinghe texture as a means of determining the stress state of the railsaken from service.

It should be noted that the textures measured in the presenttudy do not exactly represent those of the surface rail layer, buthe layer(s) a few millimeters below. This explains the accumu-ated shear strain levels being well below the aforementionedalues in the subsurface layers [33]. However, a detailed com-arison focusing on this aspect is beyond the scope of the studyresented herein.

.2. Microstructure

Many virgin and deformed samples underwent transmissionlectron microscopy (TEM) with the purpose of monitoring theicro-mechanisms responsible for the observed deformation

esponse. We do not show all the results here, as the microstruc-ural observations have not hinted at any particular mechanismesponsible for the stress–strain response exhibited. The differ-nt two-phase structures of the materials are evident, howevero planar dislocation arrangements were visible, as expected.he lack of traces of any planar dislocation activities in theicrostructure implies two possibilities: cross-slip or pencil

lide.

U

Please cite this article in press as: D. Canadinc, et al., Mater. Sci. Eng. A

When there is pencil glide, slip is possible on many systems inhe 〈1 1 1〉 direction, making it hard to observe any dislocationrrangements or traces of planar arrangements. At this point,he present observations fall short of proving whether or not

Fi

PR

OO

F

PRESSEngineering A xxx (2007) xxx–xxx

encil glide is present in any of the materials under investigation.herefore, only the primary slip on the (1 1 0)〈1 1 1〉 systems wasonsidered active during the deformation of the bcc materialsnder consideration.

Even though not presented in this paper, the authors obtainedesults from similar simulations with simultaneous activationf the primary (1 1 0)〈1 1 1〉 and (1 1 2)〈1 1 1〉 slip systems, andll possible slip systems leading to pencil glide. The differenceetween these and the current observations were nothing morehan a slight change in Voce hardening parameters and texturentensities of certain orientations. The general trend in the tex-ure was preserved. Accordingly, for the sake of simplicity, thectivation of only the primary slip in BS, PS and 1CRS wasssumed. However, based on only this decision and the indiffer-nce between the simulation results, it is not correct to disregardhe possibility of pencil glide, and the influence of pencil glides left aside for future investigations.

The current model revealed useful information regarding thelip activity in the three materials of interest under shear loadingFig. 12). Following an increase (up to 4.8) throughout a rela-ively short span of shear strain (0–0.2), the average number ofctive slip systems per grain decreases and saturates (at around.5) at larger shear strains (1.5–2.0). This observation agreesell with the expected texture evolution. As the shear deforma-

ion progresses, although no single crystallographic orientationecomes prominent, the texture evolves such that majority of therains assume orientations in the vicinity of the (1 1 1)–(1 0 1)oundary (Fig. 8). The decreased number of preferred crystal-ographic orientations limits the average number of active slipystems at large strains, which brings about the decrease from.8 to 3.5 slip systems per grain. The saturation this numbereaches, however, is expected, as the grains cannot assume a cer-ain orientation over the others due to continuous rotation of therains throughout the torsional deformation. Although individu-lly some grains might assume a single or double slip orientation,ost of the orientations assume multiple-slip orientations in the

D space (Fig. 8).

.3. Modeling of the stress–strain response

(2007), doi:10.1016/j.msea.2007.08.049

ig. 12. Average number of active slip systems (per grain) with respect tonelastic shear strain for BS, PS, and 1CRS.

D

IN+ModelM

ce and

t905

p906

d907

t908

u909

w910

r911

o912

w913

914

a915

b916

e917

r918

l919

l920

1921

i922

s923

c924

s925

i926

o927

v928

c929

o930

l931

c932

l933

h934

o935

c936

i937

s938

h939

c940

8941

942

i943

p944

e945

e946

s947

a948

i949

t950

o951

a952

B953

t954

h955

f956

E957

i958

i959

h 960

i 961

t 962

m 963

i 964

F 965

l 966

b 967

d 968

o 969

A 970

971

c 972

( 973

T 974

F 975

l 976

D 977

a 978

o 979

e 980

R 981

982

983

984

985

986

987

988

989

990

991

992

993

[ 994

995

[ 996

[ 997

[ 998

[ 999

1000

[ 1001

[ 1002

1003

[ 1004

1005

[ 1006

1007

[ 1008

1009

[ 1010

1011

[ 1012

[ 1013

NC

OR

RE

CTE

ARTICLESA 23429 1–14

D. Canadinc et al. / Materials Scien

he materials. The plastic deformation was considered as inde-endent of strain rate and temperature, and room temperatureeformation response at slow strain rates was considered inhis study. The strain rates and temperature experienced by railsnder deformation can vary; however, the purpose of this studyas to develop a methodology for predicting the deformation

esponse and texture evolution of rail materials in shear, basedn uniaxial tensile deformation response. Therefore, the focusas not placed on strain rate and temperature dependencies.Two unique strain hardening formulations were successfully

dopted to account for the important microstructural differencesetween the materials investigated. Specifically, a strain hard-ning formulation previously utilized to predict the deformationesponse of austenitic manganese steels that feature dense dis-ocation walls [14] was further modified to model the pearliteamella–glide dislocation interaction in pearlitic steels (PS andCRS). As for the bainitic microstructure (BS), the correspond-ng deformation response was modeled by adopting a modifiedtrain hardening scheme that successfully incorporates the pre-ipitate hardening into the VPSC [20]. For both cases, however,ubstantial differences exist between the modified and the orig-nal Voce hardening formulations. For instance, in the casef dense dislocation walls (HDDWs), the HDDWs evolve inolume fraction due to continuous interaction with glide dislo-ations, such that the blocked dislocations add to the densityf the HDDWs. In the current model, however, the pearliteamellae do not evolve in volume fraction, and therefore, theorresponding volume fraction was fixed throughout the simu-ations. In the case of bainitic steel simulations, on the otherand, the difference was more substantial. In particular, theriginal model not only considers the contribution due to pre-ipitate hardening, but also incorporates the twin-precipitatenteractions that are prominent in nitrogen-alloyed Hadfieldteels [20]. In the current approach, however, only the extraardening contribution due to particle hardening was taken intoonsideration.

. Conclusions

A crystal plasticity model was developed for the character-zation of the body-centered cubic (bcc) railroad steels withearlitic and bainitic microstructures, based on a combinedxperimental and simulation effort. Two unique strain hard-ning formulations were proposed that incorporate the lengthcales associated with pearlitic and bainitic microstructures intovisco-plastic self-consistent (VPSC) algorithm. Simple uniax-

al tension test data and crystallographic texture were employedo establish the parameters defining the deformation responsef the materials. The simulation of the simple shear (torsion)nd compressive deformation modes were predicted favorably.y relating the macroscopic tensile deformation response to

he deformation at the grain level based on slip systems, theardening parameters were established that led to the success-

U

Please cite this article in press as: D. Canadinc, et al., Mater. Sci. Eng. A (

ul prediction of the yield strength in shear and its evolution.xtension of our simulations to strains beyond those observed

n the laboratory but encountered in service predicted the changen texture intensities due to plastic deformation. Texture at

[[[[

PR

OO

F

PRESSEngineering A xxx (2007) xxx–xxx 13

ypothetically high strains was used as a means of determin-ng the stress state of the rails taken from service. Finally,he texture results confirm the need for developing numerical

ethods, such as the one presented herein, in order to acquirenformation that is not available from laboratory experiments.urthermore, the incorporation of length scales associated with

amellar spacing and particle hardening due to carbides in theainite structure into crystal plasticity models, as successfullyemonstrated in the current work, will assist new material devel-pment efforts.

cknowledgements

This work was supported by the Association of Ameri-an Railroads (AAR)/Transportation Technology Center Inc.TTCI) and the Federal Railway Administration (FRA).he German part of the study was supported by Deutscheorschungsgemeinschaft (DFG) within the Transregional Col-

aborative Research Center TRR30. Mr. Joe Kristan and Mr.avid Davis of AAR/TTCI are thanked for their support and

ssistance. The authors are grateful to Dr. Carlos Tome for kindlyffering the Code VPSC Version 5.0 to be used in the modelingffort presented in this paper.

eferences

[1] K. Sawley, J. Kristan, Fatigue Fract. Eng. Mater. Struct. 26 (2003)1019.

[2] K. Sawley, R. Jimenez, AAR Research Report No. R-941, 2000.[3] K.M. Lee, A.A. Polycarpou, Wear 259 (2005) 391.[4] Courtesy of AAR/TTCI.[5] K.S. Havner, Finite Plastic Deformation of Crystalline Solids, Cambridge

University Press, New York, 1992.[6] J. Weertman, J.R. Weertman, Elementary Dislocation Theory, Oxford Uni-

versity Press, New York, 1992.[7] G.I. Taylor, C.F. Elam, Proc. Roy. Soc. Lond. A 112 (1926) 337.[8] G.I. Taylor, Proc. Roy. Soc. Lond. A 118 (1928) 1.[9] C.F. Elam, Proc. Roy. Soc. Lond. A 153 (1936) 273.10] U.F. Kocks, C.N. Tome, H.R. Wenk, Texture and Anisotropy, Cambridge

University Press, New York, 1998.11] A.D. Rollett, U.F. Kocks, ICOTOM 8 (1987) 375.12] P. Gilormini, B. Bacroix, J.J. Jonas, ICOTOM 8 (1987) 381.13] P. Van Houtte, ICOTOM 5 (1978) 347.14] D. Canadinc, H. Sehitoglu, H.J. Maier, Y.I. Chumlyakov, Acta Mater. 53

(2005) 1831.15] R.A. Lebensohn, C.N. Tome, Acta Metall. Mater. 41 (1993) 2611.16] R.A. Lebensohn, C.N. Tome, Manual for Code Visco-Plastic Self-

Consistent, 2002.17] P.A. Turner, C.N. Tome, N. Christodoulou, C.H. Woo, Philos. Mag. A 79

(1999) 2505.18] D. Canadinc, H. Sehitoglu, H.J. Maier, D. Niklasch, Y.I. Chumlyakov, Int.

J. Solids Struct. 44 (2007) 34.19] D. Canadinc, H. Sehitoglu, H.J. Maier, Mater. Sci. Eng. A 454–455 (2007)

662.20] D. Canadinc, I. Karaman, H. Sehitoglu, Y.I. Chumlyakov, H.J. Maier,

Metall. Mater. Trans. A 34 (2003) 1821.21] P. Bate, W.T. Roberts, D.V. Wilson, Acta Metall. Mater. 29 (1981) 1797.22] L.M. Brown, G.R. Woolhous, Philos. Mag. 21 (1970) 329.

2007), doi:10.1016/j.msea.2007.08.049

23] L.M. Brown, W.M. Stobbs, Philos. Mag. 23 (1971) 1185. 1014

24] L.M. Brown, W.M. Stobbs, Philos. Mag. 23 (1971) 1201. 1015

25] F. Barlat, J. Liu, Mater. Sci. Eng. A 257 (1998) 47. 1016

26] T. Mura, Micromechanics of Defects in Solids, Kluwer Academic Publi- 1017

cation, 1993. 1018

IN+ModelM

1 ce and

[1019

1020

[1021

[1022

ARTICLESA 23429 1–14

4 D. Canadinc et al. / Materials Scien

UN

CO

RR

EC

TED

Please cite this article in press as: D. Canadinc, et al., Mater. Sci. Eng. A

27] E. Nembach, Particle Strengthening of Metals and Alloys, John Wiley &Sons Inc., 1997.

28] W.A. Spitzig, Acta Metall. 29 (1981) 1359.29] W.F. Hosford, The Mechanics of Crystals and Textured Polycrystals,

Oxford University Press, New York, 1993.

[[[

[

PRESSEngineering A xxx (2007) xxx–xxx

PR

OO

F

(2007), doi:10.1016/j.msea.2007.08.049

30] J.F.W. Bishop, R. Hill, Philos. Mag. 42 (1951) 414. 1023

31] D. Canadinc, H. Sehitoglu, Unpublished work. 1024

32] J. Lubliner, Plasticity Theory, Macmillan Publishing Company, USA, 1025

1990. 1026

33] F.J. Franklin, I. Widiyarta, A. Kapoor, Wear 2110 (251) 949. Q2 1027